Two sources with intensity `4I_0` , and `9I_0` , interfere at a point in medium. The minimum intensity would be

Options

A. `5 I_0`
B. `4 I_0`
C. `2 I_0`
D. `I_0`

Answer

To find the minimum intensity when two sources with intensities \(4I_0\) and \(9I_0\) interfere, we can use the formula for the net intensity due to interference:

\[ I_{\text{net}} = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos \phi \]

where:
- \(I_1 = 4I_0\)
- \(I_2 = 9I_0\)
- \(\phi\) is the phase difference between the two sources.

### Step 1: Substitute the intensities into the formula

Substituting \(I_1\) and \(I_2\) into the equation, we have:

\[ I_{\text{net}} = 4I_0 + 9I_0 + 2\sqrt{(4I_0)(9I_0)} \cos \phi \]

### Step 2: Simplify the equation

Now, simplify the equation:

\[ I_{\text{net}} = 13I_0 + 2\sqrt{36I_0^2} \cos \phi \]

### Step 3: Calculate the square root

Calculate the square root:

\[ \sqrt{36I_0^2} = 6I_0 \]

### Step 4: Substitute the square root back into the equation

Now substitute this back into the equation:

\[ I_{\text{net}} = 13I_0 + 12I_0 \cos \phi \]

### Step 5: Find the minimum intensity

To find the minimum intensity, we need to consider the minimum value of \(\cos \phi\), which is \(-1\):

\[ I_{\text{min}} = 13I_0 + 12I_0(-1) \]

### Step 6: Simplify to find the minimum intensity

This simplifies to:

\[ I_{\text{min}} = 13I_0 - 12I_0 = I_0 \]

Thus, the minimum intensity is:

\[ \boxed{I_0} \]